Lecture Notes on Applied Control Theory – "Bending moment-based force control of flexible arm under gravity" – Description of the problem
Lecture Notes on Applied Control Theory – "Bending moment-based force control of flexible arm under gravity" – Description of the problem
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Lecture Notes on Applied Control Theory
"Bending moment-based force control of flexible arm under gravity"
Luca Di Stasio, Engineer, Professional License (Italy, 2013)
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Abstract
The closed-loop system presented in the paper Bending moment-based force control of flexible arm under gravity is explained and derived in detail. Assumptions are stated clearly and references to background knowledge and facts are presented.
Keywords
flexible arm, Euler-Bernoulli beam, estimation, control, stability
References
Takahiro Endo and Haruhisa Kawasaki; Bending moment-based force control of flexible arm under gravity. Mechanism and Machine Theory 79 (2014) 217 - 229.
Fumitoshi Matsuno and Shozaburo Kasai; Modeling and robust control of constrained one-Link flexible arms. Journal of Robotic Systems 15(8), 447 - 464 (1998).
Takahiro Endo and Fumitoshi Matsuno; Force control and exponential stability for one-link flexible arm. Proceedings of the IFAC World Congress, paper code Tu-E18-To/5, (CD-ROM), 2005.
Introduction
A real-life example
Robotic surgical arm. Retrieved from www.alchetron.com on August 26, 2016.
Formulation of the model
Physical Model
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
- The arm is modeled as a slender flexible beam
- The arm is clamped to the vertical shaft of the motor
- Environment interaction is modeled by the contact with a generic object at the other end of the beam
- A concentrated mass at the other end of the beam represents the end manipulator (i.e. robotic hand)
- Gravity is present
- The system is controlled by the force $u\left(t\right)$
Assumptions
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
- 2D motion in the $x-y$ plane
- Euler-Bernoulli beam theory
- Small angles and small displacements
- The link between arm and shaft is perfectly rigid and smooth (no friction)
- No contact friction between end mass and body
Variables & Parameters
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
Reference frames
- $\mathbf{I},\mathbf{J}$: directors (unit vectors) of global inertial reference frame
- $\mathbf{i},\mathbf{j}$: directors (unit vectors) of local dragged reference frame
- $X,Y$: coordinates of global inertial reference frame
- $x,y$: coordinates of local dragged reference frame
Variables & Parameters
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
System properties
- $\rho,L,EI$: respectively density per unit length, length and flexural rigidity of the arm
- $M$: end-tip concentrated mass
- $J,\tau_{m}$: respectively the moment of inertia and the torque generated by the rotor
- $g$: acceleration of gravity at sea-level, assumed uniform
Variables & Parameters
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
Independent variables
- $\theta\left(t\right),w\left(x,t\right)$: respectively the angle of rotation of the rotor and the beam transversal displacement
- $u\left(t\right)$: control force
Kinematics
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
\[
\begin{aligned}
\mathbf{q}=\underline{\underline{R}}\mathbf{Q}\qquad&\underline{\underline{R}}=
\begin{bmatrix}
\cos{\theta} \sin{\theta}\\
-\sin{\theta} \cos{\theta}\\
\end{bmatrix}\\
\mathbf{Q}=\underline{\underline{R}}^{-1}\mathbf{q}\qquad&\underline{\underline{R}}^{-1}=
\begin{bmatrix}
\cos{\theta} -\sin{\theta}\\
\sin{\theta} \cos{\theta}\\
\end{bmatrix}\\
\end{aligned}
\]
Kinematics
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
\[
\begin{aligned}
\mathbf{\dot{i}}=\frac{d\mathbf{i}}{dt}&=\mathbf{\omega}\times\mathbf{i}=\\
&=\dot{\theta}\left(t\right)\mathbf{k}\times\mathbf{i}=\dot{\theta}\left(t\right)\mathbf{j}\\
\mathbf{\dot{j}}=\frac{d\mathbf{j}}{dt}&=\mathbf{\omega}\times\mathbf{j}=\\
&=\dot{\theta}\left(t\right)\mathbf{k}\times\mathbf{j}=-\dot{\theta}\left(t\right)\mathbf{i}\\
\end{aligned}
\]
Kinematics
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
\[
\begin{aligned}
\mathbf{r\left(x,t\right)}&=x\mathbf{i}-w\left(x,t\right)\mathbf{j}\\
\mathbf{P\left(L,t\right)}&=\mathbf{r\left(L,t\right)}=L\mathbf{i}-w\left(L,t\right)\mathbf{j}
\end{aligned}
\]
\[
\begin{aligned}
\mathbf{\dot{r}\left(x,t\right)}=&\dot{x}\mathbf{i}+x\mathbf{\dot{i}}-\dot{w}\left(x,t\right)\mathbf{j}-w\left(x,t\right)\mathbf{\dot{i}}=\\
=&x\dot{\theta}\left(t\right)\mathbf{j}-\dot{w}\left(x,t\right)\mathbf{j}+w\left(x,t\right)\dot{\theta}\left(t\right)\mathbf{j}=\\
=&\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)\mathbf{j}+w\left(x,t\right)\dot{\theta}\left(t\right)\mathbf{i}\\
\end{aligned}
\]
\[
\begin{aligned}
\mathbf{\dot{P}\left(t\right)}=&\mathbf{\dot{r}\left(L,t\right)}=\\
=&\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)\mathbf{j}+w\left(L,t\right)\dot{\theta}\left(t\right)\mathbf{i}
\end{aligned}
\]
Boundary conditions
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
\[
w\left(0,t\right)=0\qquad\dot{w}\left(0,t\right)=0
\]
Constraint surface
One-link flexible armSchematic of one-link flexible armobjectY,Jx,X,y,iIjgOw(x,t)θ(t)r(x,t)u(t)
\[
\Phi\left(X,Y\right)=Y=0
\]
\[
\begin{aligned}
X&=L\cos{\theta\left(t\right)+w\left(L,t\right)\sin{\theta\left(t\right)}}\\
Y&=L\sin{\theta\left(t\right)-w\left(L,t\right)\cos{\theta\left(t\right)}}
\end{aligned}
\]
\[
\begin{aligned}
\phi\left(\theta\left(t\right),w\left(L,t\right)\right)&=\mathbf{P}\cdot\mathbf{n}=\mathbf{P}\cdot\mathbf{\nabla}\Phi=0\\
\end{aligned}
\]
\[
\begin{aligned}
\phi\left(\theta\left(t\right),w\left(L,t\right)\right)&=X\frac{\partial\Phi}{\partial X}+Y\frac{\partial\Phi}{\partial Y}=\\
&=L\sin{\theta\left(t\right)-w\left(L,t\right)\cos{\theta\left(t\right)}}
\end{aligned}
\]
\[
Q\left(t\right)=0
\]
Derivation of governing equations in Hamiltonian form
Hamilton's principle
\[
\begin{aligned}
&\delta I=0\\
\end{aligned}
\]
\[
I=\int_{t_{1}}^{t_{2}}\mathcal{L}^{ext}dt
\]
\[
\mathcal{L}^{ext}=\mathcal{L} + W + \mathbf{\lambda}^{T}\mathbf{\phi}
\]
\[
\mathcal{L}=T-\Pi
\]
Kinetic energy
\[
T=\frac{1}{2}\int_{0}^{L}\rho\mathbf{\dot{r}}^{T}\mathbf{\dot{r}}dx+\frac{1}{2}M\mathbf{\dot{P}}^{T}\mathbf{\dot{P}}+\frac{1}{2}J\dot{\theta}^{2}=
\]
\[
\begin{aligned}
=&\frac{1}{2}\int_{0}^{L}\rho\begin{bmatrix}w\left(x,t\right)\dot{\theta}\left(t\right)&\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)\end{bmatrix}\begin{bmatrix}w\left(x,t\right)\dot{\theta}\left(t\right)\\ \left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)\end{bmatrix}dx+\\
&+\frac{1}{2}M\begin{bmatrix}w\left(L,t\right)\dot{\theta}\left(t\right)&\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)\end{bmatrix}\begin{bmatrix}w\left(L,t\right)\dot{\theta}\left(t\right)\\ \left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)\end{bmatrix}+\\
&+\frac{1}{2}J\dot{\theta}\left(t\right)^{2}=
\end{aligned}
\]
\[
\begin{aligned}
=&\frac{1}{2}\int_{0}^{L}\rho\left(\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)^{2}+w^{2}\left(x,t\right)\dot{\theta}^{2}\left(t\right)\right)dx+\\
&+\frac{1}{2}M\left(\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)^{2}+w^{2}\left(L,t\right)\dot{\theta}^{2}\left(t\right)\right)+\\
&+\frac{1}{2}J\dot{\theta}\left(t\right)^{2}
\end{aligned}
\]
Kinetic energy
\[
T=\frac{1}{2}\int_{0}^{L}\rho\mathbf{\dot{r}}^{T}\mathbf{\dot{r}}dx+\frac{1}{2}M\mathbf{\dot{P}}^{T}\mathbf{\dot{P}}+\frac{1}{2}J\dot{\theta}^{2}=
\]
\[
\begin{aligned}
=&\frac{1}{2}\int_{0}^{L}\rho\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)^{2}dx+\\
&+\frac{1}{2}M\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)^{2}+\\
&+\frac{1}{2}J\dot{\theta}\left(t\right)^{2}
\end{aligned}
\]
Potential energy
\[
\Pi=\frac{1}{2}\int_{0}^{L}EI\left(w^{''}\left(x,t\right)\right)^{2}dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)
\]
Work of external forces
\[
W=N_{g}\tau\left(t\right)\theta\left(t\right)
\]
Euler-Lagrange's equations
\[
\begin{aligned}
&\delta I=0\\
\end{aligned}
\]
\[
\delta\int_{t_{1}}^{t_{2}}\mathcal{L}^{ext}dt=0
\]
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left(\frac{\partial\mathcal{L}^{ext}}{\partial q_{i}}-\frac{d}{dt}\frac{\partial\mathcal{L}^{ext}}{\partial\dot{q}_{i}}\right)dt=0
\]
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left[\left(\frac{\partial T}{\partial q_{i}}-\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{i}}\right)-\left(\frac{\partial\Pi}{\partial q_{i}}-\frac{d}{dt}\frac{\partial\Pi}{\partial\dot{q}_{i}}\right)+\left(\frac{\partial W}{\partial q_{i}}-\frac{d}{dt}\frac{\partial W}{\partial\dot{q}_{i}}\right)+\left(\frac{\partial\left(\lambda\phi\right)}{\partial q_{i}}-\frac{d}{dt}\frac{\partial\left(\lambda\phi\right)}{\partial\dot{q}_{i}}\right)\right]dt=0
\]
Euler-Lagrange's equations
\[
\begin{aligned}
T=T\left(\dot{q}_{i}\right)&\qquad
\Pi=\Pi\left(q_{i}\right)\\
W=W\left(q_{i}\right)&\qquad
\phi=\phi\left(q_{i}\right)\\
\end{aligned}
\]
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left[-\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{i}}-\frac{\partial\Pi}{\partial q_{i}}+\frac{\partial W}{\partial q_{i}}+\lambda\frac{\partial\phi}{\partial q_{i}}\right]dt=0
\]
\[
\mathbf{q}=\begin{bmatrix}
\theta\left(t\right)\\
w\left(x,t\right)\\
w\left(L,t\right)\\
w^{'}\left(L,t\right)
\end{bmatrix}
\]
Kinetic energy variation
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left(-\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{i}}\right)dt=
\]
\[
\begin{aligned}
=&\int_{t_{1}}^{t_{2}}\left[\delta\theta\left(-\frac{d}{dt}\frac{\partial T}{\partial\dot{\theta}}\right)+\delta w\left(-\frac{d}{dt}\frac{\partial T}{\partial\dot{w}}\right)+\delta w_{e}\left(-\frac{d}{dt}\frac{\partial T}{\partial\dot{w}_{e}}\right)+\delta w^{'}_{e}\left(-\frac{d}{dt}\frac{\partial T}{\partial\dot{w}^{'}_{e}}\right)\right]dt\\&\text{with}\\
&T=\frac{1}{2}\int_{0}^{L}\rho\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)^{2}dx+\frac{1}{2}M\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)^{2}+\frac{1}{2}J\dot{\theta}\left(t\right)^{2}
\end{aligned}
\]
Kinetic energy variation
\[
\begin{aligned}
T&=\frac{1}{2}\int_{0}^{L}\rho\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)^{2}dx+\frac{1}{2}M\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)^{2}+\frac{1}{2}J\dot{\theta}\left(t\right)^{2}\\
\frac{\partial T}{\partial\dot{\theta}}&=\int_{0}^{L}\rho\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)xdx+ML\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)+J\dot{\theta}\left(t\right)\\
\frac{\partial T}{\partial\dot{w}}&=-\int_{0}^{L}\rho\left(x\dot{\theta}\left(t\right)-\dot{w}\left(x,t\right)\right)dx\\
\frac{\partial T}{\partial\dot{w}_{e}}&=-M\left(L\dot{\theta}\left(t\right)-\dot{w}\left(L,t\right)\right)\\
\frac{\partial T}{\partial\dot{w}^{'}_{e}}&=0
\end{aligned}
\]
Kinetic energy variation
\[
\begin{aligned}
\frac{d}{dt}\frac{\partial T}{\partial\dot{\theta}}&=\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+J\ddot{\theta}\left(t\right)\\
\frac{d}{dt}\frac{\partial T}{\partial\dot{w}}&=-\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)dx\\
\frac{d}{dt}\frac{\partial T}{\partial\dot{w}_{e}}&=-M\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)\\
\frac{d}{dt}\frac{\partial T}{\partial\dot{w}^{'}_{e}}&=0
\end{aligned}
\]
Kinetic energy variation
\[
\begin{aligned}
\delta T =& -\delta\theta\left[\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+J\ddot{\theta}\left(t\right)\right]+\\
&+\delta w\left[\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)dx\right]+\delta w_{e}\left[M\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)\right]
\end{aligned}
\]
Integration by parts
\[
\begin{aligned}
\Pi&=\frac{1}{2}\int_{0}^{L}EI\left(w^{''}\left(x,t\right)\right)^{2}dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)=\\
&=\frac{1}{2}EI\left[w^{''}\left(x,t\right)w^{'}\left(x,t\right)\right]_{0}^{L}-\frac{1}{2}\int_{0}^{L}EIw^{'''}\left(x,t\right)w^{'}\left(x,t\right)dx+\\+&\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)=\\
&=\frac{1}{2}EI\left[w^{''}\left(x,t\right)w^{'}\left(x,t\right)\right]_{0}^{L}-\frac{1}{2}EI\left[w^{'''}\left(x,t\right)w\left(x,t\right)\right]_{0}^{L}+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\\+&\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)=\\
&=\frac{1}{2}EI\left[w^{''}\left(L,t\right)w^{'}\left(L,t\right)-w^{''}\left(0,t\right)w^{'}\left(0,t\right)\right]-\frac{1}{2}EI\left[w^{'''}\left(L,t\right)w\left(L,t\right)-w^{'''}\left(0,t\right)w\left(0,t\right)\right]+\\&+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)
\end{aligned}
\]
Integration by parts
\[
\begin{aligned}
\Pi&=\frac{1}{2}\int_{0}^{L}EI\left(w^{''}\left(x,t\right)\right)^{2}dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)=\\
&=\frac{1}{2}EI\left[w^{''}\left(L,t\right)w^{'}\left(L,t\right)-w^{''}\left(0,t\right)w^{'}\left(0,t\right)\right]-\frac{1}{2}EI\left[w^{'''}\left(L,t\right)w\left(L,t\right)-w^{'''}\left(0,t\right)w\left(0,t\right)\right]+\\&+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)=\\
&=\frac{1}{2}EIw^{''}\left(L,t\right)w^{'}\left(L,t\right)-\frac{1}{2}EIw^{'''}\left(L,t\right)w\left(L,t\right)+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\\
+&\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)\\
\end{aligned}
\]
Potential energy variation
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left(-\frac{\partial\Pi}{\partial q_{i}}\right)dt=
\]
\[
\begin{aligned}
=&\int_{t_{1}}^{t_{2}}\left[\delta\theta\left(-\frac{\partial\Pi}{\partial\theta}\right)+\delta w\left(-\frac{\partial\Pi}{\partial w}\right)+\delta w_{e}\left(-\frac{\partial\Pi}{\partial w_{e}}\right)+\delta w^{'}_{e}\left(-\frac{\partial\Pi}{\partial w^{'}_{e}}\right)\right]dt\\&\text{with}\\
\Pi&=\frac{1}{2}EIw^{''}\left(L,t\right)w^{'}\left(L,t\right)-\frac{1}{2}EIw^{'''}\left(L,t\right)w\left(L,t\right)+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\\
+&\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)\\
\end{aligned}
\]
Potential energy variation
\[
\begin{aligned}
\Pi&=\frac{1}{2}w^{''}\left(L,t\right)w^{'}\left(L,t\right)-\frac{1}{2}w^{'''}\left(L,t\right)w\left(L,t\right)+\frac{1}{2}\int_{0}^{L}EIw^{IV}\left(x,t\right)w\left(x,t\right)dx+\\
+&\int_{0}^{L}\rho g\cos{\theta\left(t\right)}w\left(x,t\right)dx+Mg\cos{\theta\left(t\right)}w\left(L,t\right)\\
\frac{\partial\Pi}{\partial\theta}&=-\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}\\
\frac{\partial\Pi}{\partial w}&=\int_{0}^{L}EIw^{IV}\left(x,t\right)dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}dx\\
\frac{\partial\Pi}{\partial w_{e}}&=-EIw^{'''}\left(L,t\right)+Mg\cos{\theta\left(t\right)}\\
\frac{\partial\Pi}{\partial w^{'}_{e}}&=EIw^{''}\left(L,t\right)
\end{aligned}
\]
Potential energy variation
\[
\begin{aligned}
\delta\Pi =& \delta\theta\left[\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx+Mg\sin{\theta\left(t\right)w\left(x,t\right)}\right]+\\
&-\delta w\left[\int_{0}^{L}EIw^{IV}\left(x,t\right)dx+\int_{0}^{L}\rho g\cos{\theta\left(t\right)}dx\right]+\\&-\delta w_{e}\left[-EIw^{'''}\left(L,t\right)+Mg\cos{\theta\left(t\right)}\right]-\delta w_{e}^{'}\left[EIw^{''}\left(L,t\right)\right]
\end{aligned}
\]
Virtual work
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left(\frac{\partial W}{\partial q_{i}}\right)dt=
\]
\[
\begin{aligned}
=&\int_{t_{1}}^{t_{2}}\left[\delta\theta\left(\frac{\partial W}{\partial\theta}\right)+\delta w\left(\frac{\partial W}{\partial w}\right)+\delta w_{e}\left(\frac{\partial W}{\partial w_{e}}\right)+\delta w^{'}_{e}\left(\frac{\partial W}{\partial w^{'}_{e}}\right)\right]dt\\&\text{with}\\
W&=N_{g}\tau\left(t\right)\theta\left(t\right)\\
\end{aligned}
\]
Virtual work
\[
\begin{aligned}
W&=N_{g}\tau\left(t\right)\theta\left(t\right)\\
\frac{\partial W}{\partial\theta}&=N_{g}\tau\left(t\right)\\
\frac{\partial W}{\partial w}&=0\\
\frac{\partial W}{\partial w_{e}}&=0\\
\frac{\partial W}{\partial w^{'}_{e}}&=0
\end{aligned}
\]
Virtual work
\[
\delta W = \delta\theta N_{g}\tau\left(t\right)
\]
Lagrange's multipliers
\[
\int_{t_{1}}^{t_{2}}\sum_{i=0}^{n_{dof}}\delta q_{i}\left(\lambda\frac{\partial\phi}{\partial q_{i}}\right)dt=
\]
\[
\begin{aligned}
=&\int_{t_{1}}^{t_{2}}\left[\delta\theta\left(\lambda\frac{\partial\phi}{\partial\theta}\right)+\delta w\left(\lambda\frac{\partial\phi}{\partial w}\right)+\delta w_{e}\left(\lambda\frac{\partial\phi}{\partial w_{e}}\right)+\delta w^{'}_{e}\left(\lambda\frac{\partial\phi}{\partial w^{'}_{e}}\right)\right]dt\\&\text{with}\\
\phi&=\phi\left(\theta\left(t\right),w\left(L,t\right)\right)\\
\end{aligned}
\]
Lagrange's multipliers
\[
\begin{aligned}
\phi&=\phi\left(\theta\left(t\right),w\left(L,t\right)\right)\\
\frac{\partial\phi}{\partial\theta}&=\frac{\partial\phi}{\partial\theta}\\
\frac{\partial\phi}{\partial w}&=0\\
\frac{\partial\phi}{\partial w_{e}}&=\frac{\partial\phi}{\partial w_{e}}\\
\frac{\partial\phi}{\partial w^{'}_{e}}&=0
\end{aligned}
\]
Lagrange's multipliers
\[
\lambda\partial\phi= \delta\theta\lambda\frac{\partial\phi}{\partial\theta}+\delta w_{e}\lambda\frac{\partial\phi}{\partial w_{e}}
\]
Overall variation
\[
\begin{aligned}
\delta I &=-\int_{t_{1}}^{t_{2}}\delta\theta\left[\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+J\ddot{\theta}\left(t\right)\right]dt+\\
&-\int_{t_{1}}^{t_{2}}\delta\theta\left[\int_{0}^{L}-\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}-N_{g}\tau\left(t\right)-\lambda\frac{\partial\phi}{\partial\theta}\right]dt+\\
&+\int_{t_{1}}^{t_{2}}\delta w\left[\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)dx -\int_{0}^{L}EIw^{IV}\left(x,t\right)dx-\int_{0}^{L}\rho g\cos{\theta\left(t\right)}dx\right]dt+\\
&+\int_{t_{1}}^{t_{2}}\delta w_{e}\left[ M\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)-Mg\cos{\theta\left(t\right)}+\lambda\frac{\partial\phi}{\partial w_{e}}\right]dt+\\
&+\int_{t_{1}}^{t_{2}}\delta w_{e}^{'}\left[ -EIw^{''}\left(L,t\right)\right]dt=0
\end{aligned}
\]
Overall variation
\[
\begin{aligned}
\int_{0}^{L}&\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+J\ddot{\theta}\left(t\right)+\\
&-\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}-N_{g}\tau\left(t\right)-\lambda\frac{\partial\phi}{\partial\theta}=0\\
\int_{0}^{L}&\left[\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right) -EIw^{IV}\left(x,t\right)-\rho g\cos{\theta\left(t\right)}\right]dx=0\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)-Mg\cos{\theta\left(t\right)}+\lambda\frac{\partial\phi}{\partial w_{e}}=0\\
-&\frac{1}{2}EIw^{''}\left(L,t\right)=0
\end{aligned}
\]
System's equations of motion
Hamiltonian form
\[
\begin{aligned}
\int_{0}^{L}&\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+J\ddot{\theta}\left(t\right)+\\
&-\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}-N_{g}\tau\left(t\right)-\lambda\frac{\partial\phi}{\partial\theta}=0\\
\int_{0}^{L}&\left[\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right) -EIw^{IV}\left(x,t\right)-\rho g\cos{\theta\left(t\right)}\right]dx=0\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)-Mg\cos{\theta\left(t\right)}+\lambda\frac{\partial\phi}{\partial w_{e}}=0\\
-&\frac{1}{2}EIw^{''}\left(L,t\right)=0
\end{aligned}
\]
Coordinates' transformation
\[
\begin{aligned}
X&=L\cos{\theta+w_{e}\sin{\theta}}\\
Y&=L\sin{\theta-w_{e}\cos{\theta}}
\end{aligned}
\]
\[
\begin{aligned}
\mathbf{J}_{c}&=\begin{bmatrix}
\frac{\partial X}{\partial\theta}&\frac{\partial X}{\partial w_{e}}\\
\frac{\partial Y}{\partial\theta}&\frac{\partial Y}{\partial w_{e}}
\end{bmatrix}=\\
&=\begin{bmatrix}
-L\sin{\theta}+w_{e}\cos{\theta}&\sin{\theta}\\
L\cos{\theta}+w_{e}\sin{\theta}&-\cos{\theta}
\end{bmatrix}\\
\end{aligned}
\]
Control force
\[
\mathbf{J}_{c}^{T}u\left(t\right)\mathbf{n}=\lambda\begin{bmatrix}
\frac{\partial\phi}{\partial\theta}\\
\frac{\partial\phi}{\partial w_{e}}
\end{bmatrix}
\]
\[
\begin{bmatrix}
-L\sin{\theta}+w_{e}\cos{\theta}&\sin{\theta}\\
L\cos{\theta}+w_{e}\sin{\theta}&-\cos{\theta}
\end{bmatrix}^{T}u\left(t\right)\begin{bmatrix}
0\\
1
\end{bmatrix}=\lambda\begin{bmatrix}
L\cos{\theta}+w_{e}\sin{\theta}\\
-\cos{\theta}
\end{bmatrix}
\]
\[
\begin{bmatrix}
-L\sin{\theta}+w_{e}\cos{\theta}&L\cos{\theta}+w_{e}\sin{\theta}\\
\sin{\theta}&-\cos{\theta}
\end{bmatrix}\begin{bmatrix}
0\\
1
\end{bmatrix}u\left(t\right)=\lambda\begin{bmatrix}
L\cos{\theta}+w_{e}\sin{\theta}\\
-\cos{\theta}
\end{bmatrix}
\]
\[
\begin{bmatrix}
L\cos{\theta}+w_{e}\sin{\theta}\\
-\cos{\theta}
\end{bmatrix}u\left(t\right)=\lambda\begin{bmatrix}
L\cos{\theta}+w_{e}\sin{\theta}\\
-\cos{\theta}
\end{bmatrix}
\]
\[
u\left(t\right)=\lambda
\]
Small amplitude approximation
\[
\begin{aligned}
\theta\left(t\right)&\sim o\left(t\right)\\
w\left(x,t\right)&\sim o\left(x,t\right)
\end{aligned}
\]
\[
\begin{aligned}
\cos{\theta}&\approx 1\\
\sin{\theta}&\approx\theta
\end{aligned}
\]
\[
w^{n}\theta^{m}\sim o^{n+m}\left(x,t\right)\approx 0\qquad n,m\geq 1
\]
From Hamiltonian form to control formulation
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+\\
&-\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}\\
\ddot{w}&\left(x,t\right) +\frac{EI}{\rho}w^{IV}\left(x,t\right)=x\ddot{\theta}-g\cos{\theta\left(t\right)}\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)-Mg\cos{\theta\left(t\right)}+\lambda\frac{\partial\phi}{\partial w_{e}}=0\\
w^{''}&\left(L,t\right)=0
\end{aligned}
\]
From Hamiltonian form to control formulation
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+\\
&-\int_{0}^{L}\rho g\sin{\theta\left(t\right)w\left(x,t\right)}dx-Mg\sin{\theta\left(t\right)w\left(x,t\right)}=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}\\
\ddot{w}&\left(x,t\right) +\frac{EI}{\rho}w^{IV}\left(x,t\right)=x\ddot{\theta}-g\cos{\theta\left(t\right)}\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)=u\left(t\right)\cos{\theta}+Mg\cos{\theta\left(t\right)}\\
w^{''}&\left(L,t\right)=0
\end{aligned}
\]
From Hamiltonian form to control formulation
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)=\\
&=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}\\
\ddot{w}&\left(x,t\right) +\frac{EI}{\rho}w^{IV}\left(x,t\right)=x\ddot{\theta}-g\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)=u\left(t\right)+Mg\\
w^{''}&\left(L,t\right)=0
\end{aligned}
\]
From Hamiltonian form to control formulation
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\int_{0}^{L}\rho\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)xdx+ML\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)=\\
&=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}\\
\rho&\left(x\ddot{\theta}\left(t\right)-\ddot{w}\left(x,t\right)\right)=EIw^{IV}\left(x,t\right)+\rho g\\
M&\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)=u\left(t\right)+Mg-EIw^{'''}\left(L,t\right)\\
\end{aligned}
\]
From Hamiltonian form to control formulation
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\int_{0}^{L}EIw^{IV}\left(x,t\right)xdx+\int_{0}^{L}\rho gxdx+u\left(t\right)L+MgL-EILw^{'''}\left(L,t\right)=\\
&=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}
\end{aligned}
\]
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+\left[EIw^{'''}x\right]_{0}^{L}-\int_{0}^{L}EIw^{'''}\left(x,t\right)dx+\frac{1}{2}\rho gL^{2} +u\left(t\right)L+MgL-EILw^{'''}\left(L,t\right)=\\
&=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}
\end{aligned}
\]
\[
\begin{aligned}
J&\ddot{\theta}\left(t\right)+EILw^{'''}\left(L,t\right)-\left[EIw^{''}\left(x,t\right)\right]_{0}^{L}+\frac{1}{2}\rho gL^{2} +u\left(t\right)L+MgL-EILw^{'''}\left(L,t\right)=\\
&=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}
\end{aligned}
\]
From Hamiltonian form to control formulation
\[
J\ddot{\theta}\left(t\right)-EIw^{''}\left(L,t\right)+EIw^{''}\left(0,t\right)+\frac{1}{2}\rho gL^{2} +u\left(t\right)L+MgL=N_{g}\tau\left(t\right)+\lambda\frac{\partial\phi}{\partial\theta}
\]
\[
J\ddot{\theta}\left(t\right)=N_{g}\tau\left(t\right)-EIw^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL^{2} +u\left(t\right)L+MgL\right)+\lambda\frac{\partial\phi}{\partial\theta}
\]
\[
\begin{aligned}
J\ddot{\theta}\left(t\right)=&N_{g}\tau\left(t\right)-EIw^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL^{2} +u\left(t\right)L+MgL\right)+\\&+u\left(t\right)\left(L\cos{\theta\left(t\right)+w\left(L,t\right)\sin{\theta\left(t\right)}}\right)
\end{aligned}
\]
\[
J\ddot{\theta}\left(t\right)=N_{g}\tau\left(t\right)-EIw^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL+Mg\right)L
\]
From Hamiltonian form to control formulation
\[
M\left(L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)\right)+EIw^{'''}\left(L,t\right)=u\left(t\right)+Mg
\]
\[
\Phi=L\theta\left(t\right)-w\left(L,t\right)=0
\]
\[
\ddot{\Phi}=L\ddot{\theta}\left(t\right)-\ddot{w}\left(L,t\right)=0
\]
\[
u\left(t\right)=EIw^{'''}\left(L,t\right)-Mg
\]
Control formulation
\[
\begin{aligned}
&\ddot{w}\left(x,t\right)+\frac{EI}{\rho}w^{IV}\left(x,t\right)=x\ddot{\theta}\left(t\right)-g\\
&J\ddot{\theta}\left(t\right)=N_{g}\tau\left(t\right)-EIw^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL+Mg\right)L\\
&u\left(t\right)=EIw^{'''}\left(L,t\right)-Mg\\
&w\left(0,t\right)=0\qquad w^{'}\left(0,t\right)=0\\
&w^{''}\left(L,t\right)=0
\end{aligned}
\]
Homogeneous formulation
Control formulation
\[
\begin{aligned}
&\ddot{w}\left(x,t\right)+\frac{EI}{\rho}w^{IV}\left(x,t\right)=x\ddot{\theta}\left(t\right)-g\\
&J\ddot{\theta}\left(t\right)=N_{g}\tau\left(t\right)-EIw^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL+Mg\right)L\\
&u\left(t\right)=EIw^{'''}\left(L,t\right)-Mg\\
&w\left(0,t\right)=0\qquad w^{'}\left(0,t\right)=0\\
&w^{''}\left(L,t\right)=0
\end{aligned}
\]
Change of variables
\[
\ddot{w}\left(x,t\right)-x\ddot{\theta}\left(t\right)+\frac{EI}{\rho}w^{IV}\left(x,t\right)+g=0
\]
\[
\ddot{z}\left(x,t\right)+\frac{EI}{\rho}z^{IV}\left(x,t\right)=0
\]
\[
z\left(x,t\right)=w\left(x,t\right)-x\theta\left(t\right)
\]
\[
\ddot{z}\left(x,t\right)+\frac{EI}{\rho}z^{IV}\left(x,t\right)=\ddot{w}\left(x,t\right)-x\ddot{\theta}\left(t\right)+\frac{EI}{\rho}w^{IV}\left(x,t\right)
\]
\[
z\left(x,t\right)=w\left(x,t\right)-x\theta\left(t\right)+v\left(x,t\right)\quad\text{with}\quad \frac{EI}{\rho}v^{IV}\left(x,t\right)=g
\]
Change of variables
\[
\frac{EI}{\rho}v^{IV}\left(x,t\right)=g
\]
\[
\frac{EI}{\rho}v^{'''}\left(x,t\right)=gx+c_{1}
\]
\[
\frac{EI}{\rho}v^{''}\left(x,t\right)=\frac{1}{2}gx^{2}+c_{1}x+c_{2}
\]
\[
\frac{EI}{\rho}v^{'}\left(x,t\right)=\frac{1}{6}gx^{3}+\frac{1}{2}c_{1}x^{2}+c_{2}x+c_{3}
\]
\[
\frac{EI}{\rho}v\left(x,t\right)=\frac{1}{24}gx^{4}+\frac{1}{6}c_{1}x^{3}+\frac{1}{2}c_{2}x^{2}+c_{3}x+c_{4}
\]
Change of variables
\[
\begin{aligned}
&w\left(0,t\right)=0\quad\Longleftrightarrow\quad z\left(0,t\right)=0\\
&z\left(0,t\right)=w\left(0,t\right)-0\cdot\theta\left(t\right)+v\left(0,t\right)=0\\
&v\left(0,t\right)=0
\end{aligned}
\]
\[
\begin{aligned}
&\Phi=L\theta\left(t\right)-w\left(L,t\right)=0\quad\Longleftrightarrow\quad z\left(L,t\right)=0\\
&z\left(L,t\right)=w\left(L,t\right)-L\cdot\theta\left(t\right)+v\left(L,t\right)=0\\
&v\left(L,t\right)=0
\end{aligned}
\]
\[
\begin{aligned}
&z^{''}\left(0,t\right)=w^{''}\left(0,t\right)\quad\Longleftrightarrow\quad v^{''}\left(0,t\right)=0\\
&z^{''}\left(L,t\right)=w^{''}\left(L,t\right)\quad\Longleftrightarrow\quad v^{''}\left(L,t\right)=0
\end{aligned}
\]
Change of variables
\[
v\left(0,t\right)=\frac{\rho}{EI}c_{4}=0\quad\Longleftrightarrow\quad c_{4}=0
\]
\[
v^{''}\left(0,t\right)=\frac{\rho}{EI}c_{2}=0\quad\Longleftrightarrow\quad c_{2}=0
\]
\[
v^{''}\left(L,t\right)=\frac{\rho}{EI}\left(\frac{1}{2}gL^{2}+c_{1}L\right)=0\quad\Longleftrightarrow\quad c_{1}=-\frac{1}{2}gL
\]
\[
v\left(L,t\right)=\frac{\rho}{EI}\left(\frac{1}{24}gL^{4}-\frac{1}{12}gL^{4}+c_{3}L\right)=0\quad\Longleftrightarrow\quad c_{3}=\frac{1}{24}gL^{3}
\]
Change of variables
\[
v\left(x,t\right)=\frac{\rho gx}{12EI}\left(\frac{1}{2}x^{3}- L x^{2}+\frac{1}{2} L^{3}\right)
\]
\[
z\left(x,t\right)=w\left(x,t\right)-x\theta\left(t\right)+\frac{\rho gx}{12EI}\left(\frac{1}{2}x^{3}- L x^{2}+\frac{1}{2} L^{3}\right)
\]
Change of variables
\[
\ddot{z}\left(x,t\right)=\ddot{w}\left(x,t\right)-x\ddot{\theta}\left(t\right)
\]
\[
z^{IV}\left(x,t\right)=w^{IV}\left(x,t\right)+\frac{\rho g}{EI}
\]
\[
\ddot{z}^{'}\left(0,t\right)=\frac{d}{dt^{2}}w\left(0,t\right)-\ddot{\theta}\left(t\right)=-\ddot{\theta}\left(t\right)
\]
\[
z^{'''}\left(L,t\right)=w^{'''}\left(L,t\right)+\frac{\rho gL}{2EI}\quad\Longleftrightarrow\quad EIw^{'''}\left(L,t\right)=EIz^{'''}\left(L,t\right) -\frac{\rho gL}{2}
\]
Homogeneous system
\[
\begin{aligned}
&\ddot{z}\left(x,t\right)+\frac{EI}{\rho}z^{IV}\left(x,t\right)=0\\
&J\ddot{z}^{'}\left(0,t\right)=N_{g}\tau\left(t\right)-EIz^{''}\left(0,t\right)-\left(\frac{1}{2}\rho gL+Mg\right)L\\
&u\left(t\right)=EIz^{'''}\left(L,t\right)-Mg-\frac{\rho gL}{2}\\
&z\left(0,t\right)=0\qquad z\left(L,t\right)=0\\
&z^{''}\left(L,t\right)=0
\end{aligned}
\]
Controller
Control objective
\[
u\left(t\right)\rightarrow u^{d}\quad\dot{w}\left(x,t\right)\rightarrow 0\quad\dot{\theta}\left(t\right)\rightarrow 0
\]
Control objective
\[
\begin{aligned}
&u^{d}=EIz_{d}^{'''}\left(L\right)-Mg-\frac{\rho gL}{2}\\
&\frac{EI}{\rho}z_{d}^{IV}\left(x\right)=0
\end{aligned}
\]
\[
\begin{aligned}
&z_{d}^{IV}\left(x\right)=0\\
&z_{d}^{'''}\left(x\right)=c_{1}\\
&z_{d}^{''}\left(x\right)=c_{1}x+c_{2}\\
&z_{d}^{'}\left(x\right)=\frac{1}{2}c_{1}x^{2}+c_{2}x+c_{3}\\
&z_{d}\left(x\right)=\frac{1}{6}c_{1}x^{3}+\frac{1}{2}c_{2}x^{2}+c_{3}x+c_{4}\\
\end{aligned}
\]
Control objective
\[
\begin{aligned}
&u^{d}=EIz_{d}^{'''}\left(L\right)-Mg-\frac{\rho gL}{2}\\
&\frac{EI}{\rho}z_{d}^{IV}\left(x\right)=0
\end{aligned}
\]
\[
\begin{aligned}
&z_{d}^{'''}\left(L\right)=c_{1}=\frac{1}{EI}\left(u^{d}+Mg+\frac{\rho gL}{2}\right)\\
&z_{d}\left(0\right)=c_{4}=0\\
&z_{d}^{''}\left(L\right)=c_{1}L+c_{2}=0\quad\Longleftrightarrow\quad c_{2}=-c_{1}L\\
&z_{d}\left(L\right)=\frac{1}{6}c_{1}L^{3}-\frac{1}{2}c_{1}L^{3}+c_{3}L=0\quad\Longleftrightarrow\quad c_{3}=\frac{1}{3}c_{1}L^{2}\\
\end{aligned}
\]
\[
z_{d}\left(x\right)=\frac{x}{6EI}\left(u^{d}+Mg+\frac{\rho gL}{2}\right)\left(x^{2}-3Lx+2L^{2}\right)
\]
Compensator
\[
\dot{s}\left(t\right)=k_{3}\left[EIz^{''}\left(0,t\right)-s\right]
\]
Controller
\[
\tau\left(t\right)=-k_{1}EI\left[z^{''}\left(0,t\right)-z_{d}^{''}\left(0\right)\right]-k_{2}\left[EIz^{''}\left(0,t\right)-s\right]
\]
Change of variable
\[
q\left(t\right)=EIz^{''}\left(0,t\right)-s\left(t\right)
\]
\[
p\left(x,t\right)=z\left(x,t\right)-z_{d}\left(x\right)
\]
\[
x\rightarrow 1-\frac{x}{L}\qquad t\rightarrow\sqrt{\frac{EI}{L^{4}\rho}}t
\]
Closed-loop system
\[
\begin{cases}
\ddot{p}\left(x,t\right)+p^{IV}\left(x,t\right)=0\quad 0\leq x\leq 1,t\geq 0\\
p\left(0,t\right)=0\qquad p\left(1,t\right)=0\\
p^{''}\left(0,t\right)=0\\
\ddot{p}^{'}\left(1,t\right)=\tilde{k}_{1}p^{''}\left(1,t\right)+\tilde{k}_{2}q\left(t\right)\\
\dot{q}\left(t\right)=-k_{3}q\left(t\right)+\kappa\dot{p}^{''}\left(1,t\right)
\end{cases}
\]
\[
\text{with}\quad u\left(t\right)-u^{d}=-\frac{1}{L^{3}}EIp^{'''}\left(0,t\right)
\]
\[
\text{where}\quad\tilde{k}_{1}=\frac{k_{1}L^{3}\rho}{J}\quad\tilde{k}_{2}=\frac{k_{2}L^{5}\rho}{EIJ}\quad\kappa=\frac{EI}{L^{2}}\sqrt{\frac{EI}{\rho L^{4}}}
\]
Hilbert space formulation
\[
H=V\times\left(H^{2}\left(0,1\right)\cap H_{0}\left(0,1\right)\right)\times C
\]
\[
\lt z,\tilde{z}\gt_{H}=\frac{1}{2}\int_{0}^{1}u^{IV}\overline{\tilde{z}^{IV}}dx+\frac{1}{2}u^{''}\left(1\right)\overline{\tilde{u}^{''}\left(1\right)}+\frac{1}{2}\int_{0}^{1}v^{''}\overline{\tilde{v}^{''}}dx+\frac{1}{2\kappa}q\overline{\tilde{q}}
\]
\[
z=\begin{bmatrix}u\\v\\\zeta\end{bmatrix}\qquad\tilde{z}=\begin{bmatrix}\tilde{u}\\\tilde{v}\\\tilde{\zeta}\end{bmatrix}
\]
\[
V=\{u\in H^{4}\left(0,1\right):u\left(0\right)=u\left(1\right)=u^{''}\left(0\right)=0\}
\]
\[
H^{m}\left(0,1\right)=\{u\in L^{2}\left(0,1\right);D^{\alpha}u\in L^{2}\left(0,1\right)\quad\forall\alpha\in\mathcal{N}:|\alpha|\leq m\}
\]
\[
H_{0}^{1}\left(0,1\right)=\{u\in H^{1}\left(0,1\right):u\left(0\right)=u\left(1\right)=0\}
\]
Hilbert space formulation
\[
D\left(A\right)=\{z\in\left(H^{6}\left(0,1\right)\cap V\right)\times V\times C:u^{IV}\left(0\right)=u^{IV}\left(1\right)=0,\ddot{u}^{'}\left(1\right)=k_{1}u^{''}\left(1\right)+k_{2}\zeta\}
\]
\[
Az=\begin{bmatrix}v\\-u^{IV}\\-k_{3}\zeta+\kappa v^{''}\left(1\right)\end{bmatrix}
\]
First-order evolution equation
\[
\dot{z}=Az
\]
\[
z=\begin{bmatrix}p\\\dot{p}\\q\end{bmatrix}
\]
Lecture Notes on Applied Control Theory
"Bending moment-based force control of flexible arm under gravity"
Luca Di Stasio, Engineer, Professional License (Italy, 2013)